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Linear Approximation and Differentials

Introduction

Newton's method used tangent lines to "point toward" a root of the function. In this section we examine and use another geometric characteristic of tangent lines:

If \(\quad \mathrm{f}\) is differentiable at a and \(\mathrm{x}\) is close to \(\mathrm{a}\),

then the tangent line \(\mathrm{L}(\mathrm{x})\) is close to \(\mathrm{f}(\mathrm{x})\). (Fig. 1)


This idea is used to approximate the values of some commonly used functions and to predict the "error" or uncertainty in a final calculation if we know the "error" or uncertainty in our original data. Finally, we define and give some examples of a related concept called the differential of a function.


Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.9-Linear-Approximation.pdf
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